For any two complex numbers z1 and z2 prove that
Re(z1z2)=Re(z1)Re(z2)−Im(z1)Im(z2)
Let z1=a1+ib1 and z2=a2+ib2
Then Re(z1)=a1, Re(z2)=a2, Im(z1)=b1
and Im(z2)=b2
Now z1z2=(a1+ib1)(a2+ib2)=a1a2+ib1b2+ia1b1+i2b1b2=(a1a2−b1b2)+i(a1b2+a2b1) (∵ i2=−1)
∴ Re(z1z2)=a1a2−b1b2
=Re(z1)Re(z2)−Im(z1)Im(z2)